Every Euler tour in a graph yields a Hamiltonian cycle in its line graph.
Though this map is not bijective we show that we can make it so by altering
the input graph slightly. It is well known [1]} that the line graphs
of bounded tree-width graphs have bounded clique-width. We show how to
obtain a bounded clique-width decomposition for the line graph of a bounded
tree-width graph in Logspace using the Logspace version of Courcelle's
Theorem [2]} by first obtaining a bounded tree-width decomposition via
a Logspace version of Bodlaender's theorem [2]}.
